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Math 060/Final Exam Review Guide/ 2010-2011/ College of the Canyons General Information:
• The final exam is a 2-hour timed exam. • There will be approximately 40 questions. • There will be no calculators or notes allowed. • You will be given the formulas for how to factor the sum and difference of cubes.
3 3 2 2( )( )a b a b a ab b− = − + + and 3 3 2 2( )( )a b a b a ab b+ = + − + Topics:
• Solve linear inequalities. (Section 2.8) • Write and graph equations of lines, including finding slope. (Sections 3.2-3.6) • Solve systems of linear equations. (Sections 4.1-4.3) • Multiply and divide polynomials. (Sections 5.2-5.5) • Simplify expressions with zero and negative exponents. (Section 5.4) • Factor polynomials. (Sections 6.1-6.5) • Add, subtract, multiply, divide rational expressions. (Sections 7.2, 7.3, 7.5, 7.6) • Solve equations. (Sections 2.3, 2.4, 6.6, 7.7) • Solve word problems. (Sections 2.6, 2.7, 4.4, 4.5, 6.7, 7.8)
Study Tip: Write one math question (from the sample final, your past exams, and the book) per 3x5 card. On the back of the 3x5 card write where you found the problem and the answer. Mix the cards up when you practice solving the problems. Write notes to yourself on the back of the cards if you need to remember formulas or other steps. Formulas to Remember:
Slope between two points: 2 1
2 1
y ymx x−
=−
Slope of parallel lines: 2 1m m=
Slope of perpendicular lines: 121 −=⋅mm or 21
1mm
= −
Slope-Intercept Form of a Line: y mx b= + An x-intercept is (x, 0). In other words, let y = 0. Point-Slope Form of a Line: 1 1( )y y m x x− = − A y-intercept is (0, y). In other words, let x = 0. Vertical lines have the equation x = a with an undefined slope. Horizontal lines have the equation y = b with a slope of zero. The graph of the lines y mx= and 0Ax By+ = go through the origin. Formulas to Remember, continued:
Exponent Rules (See page 348.) Product nmnm aaa +=⋅ Power ( ) nmnm aa ⋅= Quotient m
m nn
a aa
−= if 0a ≠
Product to a Power ( )n n nab a b= Quotient to a Power n n
n
a ab b
⎛ ⎞ =⎜ ⎟⎝ ⎠
if 0b ≠
Zero Exponent 0 1a = if 0a ≠ Negative Exponent 1n
naa
− = if 0a ≠
1 n
n aa− = if 0a ≠
Quotient to a Negative Power n na b
b a
−⎛ ⎞ ⎛ ⎞=⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
if 0a ≠ , 0b ≠
Polynomials: Special Forms
2 2
2 2 2
2 2 2
( )( )( ) 2( ) 2
A B A B A BA B A AB BA B A AB B
− + = −
− = − +
+ = + +
Uniform Motion: (Rate)(Time) = Distance OR (Distance) / (Rate) = Time If wind or current affects the rate, use x y+ for traveling “with” and use x y− for traveling “against.” The object’s speed is represented by x, and the wind (or current) speed is represented by y. Simple Interest: (Principal$)(Rate%) = Interest$ [This formula is good over a length of time = 1 year.] [Remember to change % into decimal form.] Mixture: (Quantity)(Concentration%) = Amount [Remember to change % into decimal form.]
Work: 1 1 1individual time individual time t
+ = where t represents time together.
Complementary angles add up to 90° . Supplementary angles add up to 180° .
Area of a rectangle: A LW= Area of a triangle: 12 2
bhA bh= =
Pythagorean Theorem: 2 2 2a b c+ =
Sample Problems for the Math 060 Final Exam, 2010-2011 (Sullivan, Struve, Mazzarella combo book)
1. Solve the inequalities. Graph the solution set on a number line and write the solution set in interval notation.
a. 1 2 3x− ≤ (2.8 #71)
b. 1 ( 4) 82
x x− > + (2.8 #77)
c. 4( 1) 3( 1)x x x− > − + (2.8 #79)
d. 4(2 1) 3( 2) 5( 2)w w w− ≥ + + − (2.8 #85)
2. Graph each linear equation by finding its intercepts. Write the coordinates of the x-intercept: ( , ). Write the coordinates of the y-intercept: ( , ).
a. 3 6 18x y+ = (3.2 #67)
5 –5
5
–5
b. 5 15x y− + = (3.2 #69)
c. 9 2 0x y− = (3.2 #73)
3. Graph the following lines. Label at least two points on the graph grid.
a. 5x = (3.2 #83)
b. 6y = − (3.2 #85)
5 –5
5
–5
5 –5
5
–5
5 –5
5
–5
5 –5
5
–5
4. Graph the line that contains the given point and has the given slope. Label at least two points on the
graph grid.
a. (2, 3); 0m− = (3.3 #51)
b. 2(2,1);3
m = (3.3 #53)
c. 5( 1,4);3
m− = − (3.3 #55)
d. (0,0); is undefinedm (3.3 #57)
5 –5
5
–5
5 –5
5
–5
5 –5
5
–5
5 –5
5
–5
5. Use the slope and y-intercept to graph each line. Label at least two points on the graph grid.
a. 3y x= + (3.4 #39)
b. 5 22
y x= − − (3.4 #45)
c. 3 2 10x y− = (3.4 #51)
d. 3xy = (3.4 #81)
5 –5
5
–5
5 –5
5
–5
5 –5
5
–5
5 –5
5
–5
6. Write the equation of the line. Your answer should be in slope-intercept form y mx b= + .
a. Through the points ( 3,2) and (1, 4)− − (3.5 #41)
b. Through the points ( 3, 11) and (2, 1)− − − (3.5 #43)
c. Through the points ( 2,3) and (4, 6)− − (3.5 #67)
d. Through the point (10,2) and parallel to the line 3 2 5x y− = (3.6 #51)
e. Through the point ( 1, 10)− − and parallel to the line 2 4x y+ = (3.6 #53)
f. Through the point ( 4, 1)− − and perpendicular to the line 4 1y x= − + (3.6 #57)
g. Through the point (7,5) and perpendicular to the y-axis. (3.6 #61)
7. Solve the following systems of linear equations. If there is one solution, write the solution as an ordered
pair (x,y). Otherwise if there are infinitely many solutions or no solution, state so.
a. Solve by graphing: 3 16 2 4x yx y− = −
− + = − (4.1 #31)
b. Solve by substitution or elimination: 7
2 2x yx y+ = −− = −
(4.2 #17)
c. Solve by substitution or elimination: 3 16 2 2x yx y+ = −+ = −
(4.3 #21)
d. Solve by substitution or elimination: 2 4 05 2 6
x yx y+ =+ =
(4.3 #29)
8. Find the product.
a. 2 3 3( ) ( 2 )x y xy− (5.2 #85)
b. 2 2( 3)( 1)x x+ + (5.3 #55)
c. 2( 2)x − (5.3 #73)
d. 2(5 3)k − (5.3 #75)
e. 2( 2 )x y+ (5.3 #77)
f. 2( 2)( 3 1)x x x− + + (5.3 #83)
9. Simplify. Write answers with positive exponents. (If you have a number raised to a positive exponent,
find its value.)
a. 0(24 )ab (5.4 #55)
b. 22
5
−⎛ ⎞⎜ ⎟⎝ ⎠
(5.4 #69)
c. 3
2
2nm
−⎛ ⎞−⎜ ⎟⎝ ⎠
(5.4 #73)
d. 2
14− (5.4 #75)
e. 3
52m− (5.4 #79)
f. 2 3
2 1
213
y zy z
−
− − (5.4 #87)
10. Divide and simplify.
a. 24 22
x xx− (5.5 #13)
b. 3 2
2
9 27 33
a aa
+ − (5.5 #15)
c. 4 3 10 4
2x x x
x− + −
+ (5.5 #37)
d. 3 22 7 10 5
2 1x x x
x+ − +
− (5.5 #47)
e. 3 2
2
2 82
x xx+ −
− (5.5 #53)
11. Factor completely. If the polynomial cannot be factored, say that it is prime. (In Section 6.3 the author
asks students to factor by trial and error or by grouping. This will not be the case on the final exam.
You may choose a method that works for you.)
a. 3 2 1x x x+ + + (6.5 #47)
b. 2 100x − (6.5 #19)
c. 29 a− (6.5 #33)
d. 3 212 2 2x x x− + + (6.5 #71)
e. 3 38 27x y− (6.4 #51)
f. 2 10 25x x+ + (6.4 #29)
g. 216 24 9x x+ + (6.4 #33)
h. 24 12 9z z− + (6.4 #37)
i. 327 x+ (6.4 #49)
j. 2 29x y+ (6.4 #85)
k. 22 5 3x x+ + (6.3 #23)
l. 24 11 3p p− + + (6.3 #29)
m. 26 17 10n n− + (6.3 #35)
n. 25 13 6w w+ − (6.3 #49)
o. 26 17 12x x− − (6.3 #85)
p. 2 330 22 24x x x+ − (6.3 #93)
q. 2 2 2 26 ( 1) 25 ( 1) 14( 1)x x x x x+ − + + + (6.3 #95)
r. 2 9 18m m+ + (6.2 #23)
s. 2 12 45z z+ − (6.2 #33)
t. 2 25 6x xy y− + (6.2 #35)
u. 22 8 8y y− + − (6.2 #51)
v. 3 2 44 32x x x− + (6.2 #53)
w. 2 4 21g g− + (6.2 #71)
x. 2 4 3 3 215 60 45a b ab a b− + (6.1 #63)
y. 2 2( 1) ( 1)x x y x− + − (6.1 #67)
z. 3 22 4 2t t t− − + (6.1 #77)
12. Perform the indicated operation.
a. 22 984
63
yy
yyy +
⋅−−
(7.2 #37)
b. 2362
321
2
22
++−+
⋅−−
pppp
pp (7.2 #19)
c. 2 2
2 2
9 2 35 4 4x x x
x x x x− − −
÷+ + +
(7.2 #59)
d.
2 48
22
c
c
−
− (7.2 #47)
e. 2 2
23 10 3 10a
a a a a+
− − − − (7.3 #31)
f. 2 222 2
x x x xx x− −
− (7.3 #35)
g. 2 2 2 2
2 2x yx y y x
−− −
(7.3 #55)
h. 6 4 12 6 3
aa a
−−
+ + (7.5 #41)
i. 2
3 1 99
xx x x−
−−
(7.5 #61)
j. 2 2
2 1 34 6
n nn n n
++
− − − (7.5 #67)
k. 2
2 3 102 4
mm m m
+− +
+ − (7.5 #77)
l.
3 12 45 16 2
−
+ (7.6 #25)
m.
2
2
2
16 41
16 4
b bb b
bb b
−− +
−− −
(7.6 #43)
n. 2
2
2 31
91
x x
x
− −
− (7.6 #45)
13. Solve the equations. Check your solution.
a. 14 3 2a a
− = − check: 14 3 2a a
− = − (2.3 #35)
b. 0.05 157.5p p+ = check: 0.05 157.5p p+ = (2.3 #51)
c. 0.02(2 24) 0.4( 1)c c− = − − check: 0.02(2 24) 0.4( 1)c c− = − − (2.3 #59)
14. Solve the equations.
a. Solve for r: PrA P t= + (2.4 #53)
b. Solve for b: 1 ( )2
A h B b= + (2.4 #55)
c. Solve for y: 3
x zy=
+ (7.7 #53)
d. Solve for S: 1 1 1R S T
= − (7.7 #55)
15. Solve the equations.
a. 2 9 14 0n n+ + = (6.6 #37)
b. 24 2 0x x+ = (6.6 #39)
c. 2 6n n− = (6.6 #47)
d. 3 22 2 12 0x x x+ − = (6.6 #59)
e. 3 23 4 12 0y y y+ − − = (6.6 #61)
f. 22 ( 1) 8a a a+ = + (6.6 #73)
16. Solve the equations. Remember to check for values of the variable which make the expressions in
each rational equation undefined.
a. 2 722 2x x+ =
+ + (7.7 #21)
b. 2
2 3 61 1 1a a a
−+ =
− + − (7.7 #25)
c. 32 8
xx x
=− +
(7.7 #35)
Warning: The bold printed words shown below at the beginning of the word problems will not always
appear with the word problems listed on the final.
17. Bad Investment. (2.6 #43)
After Mrs. Fisher lost 9% of her investment, she had $22,750. What was Mrs. Fisher’s original investment?
Variable and what it represents: __________________________
Equation:_____________________________
Original Investment:______________
18. Commission. (2.6 #53)
Melanie receives a 3% commission on every house she sells. If she received a commission of $8571, what
was the value of the house she sold?
Variable and what it represents: ____________________________
Equation:_____________________________
Value of House Sold:______________
19. Angles. (2.7 #13)
Find two supplementary angles such that the measure of the first angle is 10° less than three times the
measure of the second.
Variable and what it represents: _______________________________
Equation:_____________________________
Measures of both angles:______________
20. Uniform Motion. (2.7 #39)
Two boats leave a port at the same time, one going north and the other traveling south. The north-bound
boat travels 16 mph faster than the south-bound boat. If the southbound boat is traveling at 47 mph, how
long will it be before they are 1430 miles apart?
(You may use the table below to help set up your equation and solve the problem.)
× =
× =
× =
Equation: __________________________
Length of time: __________________
21. Uniform Motion. (2.7 #43)
A 360-mile trip began on a freeway in a car traveling at 62 mph. Once the road became a 2-lane highway,
the car slowed to 54 mph. If the total trip took 6 hours, find the time spent on each type of road.
(You may use the table below to help set up your equation and solve the problem.)
× =
× =
× =
Equation:____________________________
Time spent on freeway: ______________
Time spent on 2-lane highway:________________
22. Uniform Motion. (2.7 #45) Carol knows that when she jogs along her neighborhood greenway, she can complete the route in 10 minutes.
It takes 30 minutes to cover the same distance when she walks. If her jogging rate is 4 mph faster than her
walking rate, find the speed at which she jogs.
(You may use the table below to help set up your equation and solve the problem.)
× =
× =
× =
Equation:____________________________
Jogging speed:_________________
23. Uniform Motion. (4.4 #37)
Vanessa and Richie are riding their bikes down a trail to the next campground. Vanessa rides at 10 mph
while Richie rides at 7 mph. Since Vanessa is a little speedier, she stays behind and cleans up camp for 30
minutes before leaving. How long has Richie been riding when Vanessa is 7 miles ahead of Richie?
(You may use the table below to help set up your equation and solve the problem.)
× =
× =
× =
Equation:____________________________
Length of time:_______________
24. Uniform Motion. (4.4 #33)
Suppose that Jose bikes into the wind for 60 miles and it takes him 6 hours. After a long rest, he returns
(with the wind at his back) in 5 hours. Determine the speed at which Jose can ride his bike in still air and
determine the effect that the wind had on his speed.
(You may use the table below to help set up your system of equations and solve the problem.)
× =
× =
× =
System of Equations:______________________
Jose’s speed:___________
Wind speed:_________
25. Uniform Motion. (4.4 #39)
With a tailwind, a small Piper aircraft can fly 600 miles in 3 hours. Against this same wind, the Piper can fly
the same distance in 4 hours. Find the effect of the wind and the average airspeed of the Piper.
(You may use the table below to help set up your system of equations and solve the problem.)
× =
× =
× =
System of Equations:________________________
Piper’s speed:__________
Wind speed:________
26. Oakland Baseball. (4.4 #19)
The attendance at the games on two successive nights of Oakland A’s baseball was 44,000. The attendance
on Thursday’s game was 7000 more than two-thirds of the attendance at Friday night’s game. How many
people attended the baseball game each night?
Variables and what they represent: ___________________________
System of Equations:____________________________
Attendance on Thursday:____________
Attendance on Friday:____________
27. Angles. (4.4 #27)
The measure of one angle is 15°more than half the measure of its complement. Find the measures of the two
angles.
Variables and what they represent: ____________________________
System of Equations:__________________________
Measure of one angle: _________________
Measure of the other angle:_____________
28. Interest. (4.5 #29)
Harry has $10,000 to invest. He invests in two different accounts, one expected to return 5% and the other
expected to return 8%. If he wants to earn $575 for the year, how much should he invest at each rate?
(You may use the table below if you wish.)
× =
× =
× =
× =
System of Equations or Equation:__________________________
Amount invested at 5%:__________________
Amount invested at 8%:___________________
29. Mixture. (4.5 #39)
A lab technician needs 60 ml of a 50% saline solution. How many ml of 30% saline solution should she add
to a 60% saline solution to obtain the required mixture?
(You may use the table below if you wish.)
× =
× =
× =
× =
System of Equations or Equation:_________________________
Quantity of 30% solution:______________
30. Mixture. (4.5 #41)
How many liters of 10% silver must be added to 70 liters of 50% silver to make an alloy that is 30% silver?
You may use the table below if you wish.)
× =
× =
× =
× =
System of Equations or Equation:__________________
Quantity of 10% solution: _________________
31. Area of a Triangle. (6.7 #29)
The sail on a sailboat is in the shape of a triangle. If the height of the sail is 3 times the length of the base
and the area is 54 square feet, find the dimensions of the sail.
Equation:____________________
Base of sail:_________________
Height:___________________
32. Pythagorean Theorem. (6.7 #31)
Your big-screen TV measures 50 inches on the diagonal. If the front of the TV measures 40 inches across
the bottom, find the height of the TV.
Equation:____________________
Height of TV:_____________
33. Area of a Rectangle. (6.7 #33)
The length of a rectangle is 1 mm more than twice the width. If the area is 300 square mm, find the
dimensions of the rectangle.
Equation:__________________________________
Length:________________
Width:_____________
34. Uniform Motion. (7.8 #75)
While training for an iron man competition, Tony bikes for 60 miles and runs for 15 miles. If his biking
speed is 8 times his running speed and it takes 5 hours to complete the training, how long did he spend on his
bike?
÷ =
÷
÷
Equation:______________________________
Length of time: ________________
35. Uniform Motion. (7.8 #79)
You have a 20-mile commute into work. Since you leave very early, the trip going to work is easier than the
trip home. You can travel to work in the same time that it takes for you to make it 16 miles on the trip back
home. Your average speed coming home is 7 miles per hour slower than your average speed going to work.
What is your average speed going to work?
÷ =
Equation:______________________________
Average speed to work: _____________
36. Uniform Motion. (7.8 #73)
A boat can travel 12 km down the river in the same time it can go 4 km up the river. If the current in the
river is 2 km per hour, how fast can the boat travel in still water?
÷ =
Equation: _______________________
Boat’s speed: _______________
37. Work. (7.8 #65)
After hitting practice for the Long Beach State volleyball team, Dyanne can retrieve all of the balls in the
gym in 8 minutes. It takes Makini 6 minutes to retrieve all the balls. If they work together, to the nearest
tenth of a minute, how long will it take these two players to return the volleyballs and be ready to start the
next round of hitting practice?
Equation: _______________________
Time together:________________
38. Work. (7.8 #69) It takes an apprentice twice as long as the experienced plumber to replace the pipes
under an old house. If it takes them 5 hours when they work together, how long would it take the apprentice
alone?
Equation: _______________________
Time for apprentice alone:________________
39. Work. (7.8 #71) Using a single hose, Janet can fill a pool in 6 hours. The same pool can be drained in
8 hours by opening a drainpipe. If Janet forgets to close the drainpipe, how long would it take her to till the
pool?
Equation: _______________________
Time to fill pool:________________